elemprob-fall2010-page1 - A is 1 36 times the number of...

Math 3160, Lecture 31FrameworkWe start with a sample space Ω, a class of events, and a probability.The sample space can be any set whatsoever. An event is a subset of thesample space. Occasionally we restrict the collection of subsets to what iscalled aσ-field, but most often we allow any subset of Ω to be an event.To each eventAwe assign a numberP(A), the probability ofA, which is anumber between 0 and 1.As an example, suppose we roll a die.(“Die” is the singular form ofthe plural “dice.”) We set Ω ={1,2,3,4,5,6}and the set of events is thecollection of all subsets of Ω. We assume the die is fair, and we setP(A) to beequal to the number of elements inAdivided by 6. ThusP({1,3,5}) =36=12.For another example, suppose we roll two dice, one green and one red.Then we take Ω to be all pairs (1,1),(1,2), . . . ,(6,6) and the probability of

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Unformatted text preview: A is 1 / 36 times the number of elements in A . In both these cases, we assume that each number on each die is equally likely. We might have a “loaded” die, where P ( { 1 } ) = . 10, P ( { 2 } ) = . 15, etc. We do want something to happen, so P (Ω) = 1 and P ( ∅ ) = 0. The collection of events have to be what is called a σ-ﬁeld. A collection F subsets of Ω is a σ-ﬁeld if Ω , ∅ ∈ F , A c ∈ F whenever A ∈ F , and ∪ ∞ i =1 A i and ∩ ∞ i =1 A i are in F whenever all the A i are in F . A probability is a function on the events such that (1) 0 ≤ P ( A ) ≤ 1 for each event A . (2) P (Ω) = 1 and P ( ∅ ) = 0. (3) If A and B are disjoint, which means A ∩ B = ∅ , then P ( A ∪ B ) = P ( A ) + P ( B ) . 1...
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